Ballantine’s type theorem for complex symplectic group۱

In the late ۱۹۶۰ Ballantine showed that every matrix with positive determinant isa product of five positive definite matrices.We consider the complex symplectic group Sp(۲n,C):(…)The symplectic group is a classical group defined as the set of linear transformationsof a ۲n-dimensional vector space over C, which preserve the non-degenerate skewsymmetricbilinear form that is defined by Jn. We show that every symplectic matrixis a product of five positive definite symplectic matrices. We also show that five is thebest in the sense that there are symplectic matrices which are not product of less.This is a joint work with Daryl Q. Granario, De La Salle University, Philippines